There are two related works appearing very recently. The first one [1]} mainly investigated the trade-off between adversarial risk and standard risk under an isotropic condition of the covariate. Rather, we focus on how to improve adversarial robustness through utilizing prior knowledge on the model, and study statistical properties of the adversarially robust estimate itself, in contrast with the generalization studies by [2]}, [3]}, [4]}, [5]}. Another recent work [6]} studied the sharp statistical bound in adversarially robust classification. In the regression setup, our theorems reveal that an adversarially robust estimate is different from a standard estimate even in the rate of convergence: for noiseless case, standard model estimators can exactly recover the correct model, but the lower bound for adversarially robust model is always nonzero. Our lower bound for sparse model is also new.
